Mathematical Definition and Systematization of Puzzle Rules

• URL: http://arxiv.org/abs/2501.01433v2
• Date: Wed, 08 Jan 2025 13:08:32 GMT
• Title: Mathematical Definition and Systematization of Puzzle Rules
• Authors: Itsuki Maeda, Yasuhiro Inoue,
• Abstract summary: We introduce a mathematical framework for defining and systematizing pencil puzzle rules. This framework formalizes grid elements, their positional relationships, and iterative composition operations. Applying this framework, we successfully formalized the rules of well-known Nikoli puzzles, including Slitherlink and Sudoku.
• Score: 0.0
• License:
• Abstract: While logic puzzles have engaged individuals through problem-solving and critical thinking, the creation of new puzzle rules has largely relied on ad-hoc processes. Pencil puzzles, such as Slitherlink and Sudoku, represent a prominent subset of these games, celebrated for their intellectual challenges rooted in combinatorial logic and spatial reasoning. Despite extensive research into solving techniques and automated problem generation, a unified framework for systematic and scalable rule design has been lacking. Here, we introduce a mathematical framework for defining and systematizing pencil puzzle rules. This framework formalizes grid elements, their positional relationships, and iterative composition operations, allowing for the incremental construction of structures that form the basis of puzzle rules. Furthermore, we establish a formal method to describe constraints and domains for each structure, ensuring solvability and coherence. Applying this framework, we successfully formalized the rules of well-known Nikoli puzzles, including Slitherlink and Sudoku, demonstrating the formal representation of a significant portion (approximately one-fourth) of existing puzzles. These results validate the potential of the framework to systematize and innovate puzzle rule design, establishing a pathway to automated rule generation. By providing a mathematical foundation for puzzle rule creation, this framework opens avenues for computers, potentially enhanced by AI, to design novel puzzle rules tailored to player preferences, expanding the scope of puzzle diversity. Beyond its direct application to pencil puzzles, this work illustrates how mathematical frameworks can bridge recreational mathematics and algorithmic design, offering tools for broader exploration in logic-based systems, with potential applications in educational game design, personalized learning, and computational creativity.

Related papers

• Beyond Interpolation: Extrapolative Reasoning with Reinforcement Learning and Graph Neural Networks [18.982541044390384]
  This study focuses on the impact of the inductive bias of the architecture, different reward systems and the role of recurrent modeling in enabling sequential reasoning. We show how these elements contribute to successful extrapolation on increasingly complex puzzles.
  arXiv  Detail & Related papers  (2025-02-06T08:07:35Z)
• Improving Complex Reasoning over Knowledge Graph with Logic-Aware Curriculum Tuning [89.89857766491475]
  We propose a complex reasoning schema over KG upon large language models (LLMs) We augment the arbitrary first-order logical queries via binary tree decomposition to stimulate the reasoning capability of LLMs. Experiments across widely used datasets demonstrate that LACT has substantial improvements(brings an average +5.5% MRR score) over advanced methods.
  arXiv  Detail & Related papers  (2024-05-02T18:12:08Z)
• Are Language Models Puzzle Prodigies? Algorithmic Puzzles Unveil Serious Challenges in Multimodal Reasoning [24.386388107656334]
  This paper introduces the novel task of multimodal puzzle solving, framed within the context of visual question-answering. We present a new dataset, AlgoVQA, designed to challenge and evaluate the capabilities of multimodal language models in solving algorithmic puzzles.
  arXiv  Detail & Related papers  (2024-03-06T17:15:04Z)
• Multi-Phase Relaxation Labeling for Square Jigsaw Puzzle Solving [73.58829980121767]
  We present a novel method for solving square jigsaw puzzles based on global optimization. The method is fully automatic, assumes no prior information, and can handle puzzles with known or unknown piece orientation.
  arXiv  Detail & Related papers  (2023-03-26T18:53:51Z)
• Automated Graph Genetic Algorithm based Puzzle Validation for Faster Game Desig [69.02688684221265]
  This paper presents an evolutionary algorithm, empowered by expert-knowledge informeds, for solving logical puzzles in video games efficiently. We discuss multiple variations of hybrid genetic approaches for constraint satisfaction problems that allow us to find a diverse set of near-optimal solutions for puzzles.
  arXiv  Detail & Related papers  (2023-02-17T18:15:33Z)
• Are Deep Neural Networks SMARTer than Second Graders? [85.60342335636341]
  We evaluate the abstraction, deduction, and generalization abilities of neural networks in solving visuo-linguistic puzzles designed for children in the 6--8 age group. Our dataset consists of 101 unique puzzles; each puzzle comprises a picture question, and their solution needs a mix of several elementary skills, including arithmetic, algebra, and spatial reasoning. Experiments reveal that while powerful deep models offer reasonable performances on puzzles in a supervised setting, they are not better than random accuracy when analyzed for generalization.
  arXiv  Detail & Related papers  (2022-12-20T04:33:32Z)
• Video Anomaly Detection by Solving Decoupled Spatio-Temporal Jigsaw Puzzles [67.39567701983357]
  Video Anomaly Detection (VAD) is an important topic in computer vision. Motivated by the recent advances in self-supervised learning, this paper addresses VAD by solving an intuitive yet challenging pretext task. Our method outperforms state-of-the-art counterparts on three public benchmarks.
  arXiv  Detail & Related papers  (2022-07-20T19:49:32Z)
• Brick-by-Brick: Combinatorial Construction with Deep Reinforcement Learning [52.85981207514049]
  We introduce a novel formulation, complex construction, which requires a building agent to assemble unit primitives sequentially. To construct a target object, we provide incomplete knowledge about the desired target (i.e., 2D images) instead of exact and explicit information to the agent. We demonstrate that the proposed method successfully learns to construct an unseen object conditioned on a single image or multiple views of a target object.
  arXiv  Detail & Related papers  (2021-10-29T01:09:51Z)
• Using Small MUSes to Explain How to Solve Pen and Paper Puzzles [4.535832029902474]
  We present Demystify, a tool which allows puzzles to be expressed in a high-level constraint programming language. We give several improvements to the existing techniques for solving puzzles with MUSes. We demonstrate the effectiveness and generality of Demystify by comparing its results to documented strategies for solving a range of pen and paper puzzles by hand.
  arXiv  Detail & Related papers  (2021-04-30T15:07:51Z)
• Pictorial and apictorial polygonal jigsaw puzzles: The lazy caterer model, properties, and solvers [14.08706290287121]
  We formalize a new type of jigsaw puzzle where the pieces are general convex polygons generated by cutting through a global polygonal shape/image with an arbitrary number of straight cuts. We analyze the theoretical properties of such puzzles, including the inherent challenges in solving them once pieces are contaminated with geometrical noise.
  arXiv  Detail & Related papers  (2020-08-17T22:07:40Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.